A101093 Second partial sums of sixth powers (A001014).

1, 66, 860, 5750, 26265, 93436, 278256, 725220, 1703625, 3682030, 7431996, 14167946, 25730705, 44823000, 75305920, 122566056, 193963761, 299373690, 451829500, 668285310, 970507241, 1386109076, 1949746800, 2704487500

Offset

1,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Luciano Ancora, Recurrence relation for the second partial sums of m-th powers

Luciano Ancora, Second partial sums of the m-th powers

C. P. Neuman and D. I. Schonbach, Evaluation of sums of convolved powers using Bernoulli numbers, SIAM Rev. 19 (1977), no. 1, 90--99. MR0428678 (55 #1698). See Table 1. - N. J. A. Sloane, Mar 23 2014

C. Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube.

Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Formula

a(n) = n*(1 + n)^2*(2 + n)*(-1 + n*(2 + n))*(-2 + 3*n*(2 + n))/168.

G.f.: x*(1+x)*(1 + 56*x + 246*x^2 + 56*x^3 + x^4)/(1-x)^9. - Colin Barker, Dec 18 2012

a(n) = Sum_{i=1..n} i*(n+1-i)^6, by the definition. - Bruno Berselli, Jan 31 2014

a(n) = 2*a(n-1) - a(n-2) + n^6. - Luciano Ancora, Jan 08 2015

Maple

f:=n->(3*n^8-14*n^6+21*n^4-10*n^2)/168;
[seq(f(n), n=0..50)];  # N. J. A. Sloane, Mar 23 2014

Mathematica

CoefficientList[Series[(x+1)(x^4+56x^3+246x^2+56x+1)/(1-x)^9, {x, 0, 40}], x] (* Vincenzo Librandi, Mar 24 2014 *)
Nest[Accumulate, Range[30]^6, 2] (* or *) LinearRecurrence[{9, -36, 84, -126, 126, -84, 36, -9, 1}, {1, 66, 860, 5750, 26265, 93436, 278256, 725220, 1703625}, 30] (* Harvey P. Dale, Jun 05 2019 *)

Prog

(Magma) [n*(1+n)^2*(2+n)*(-1+n*(2+n))*(-2+3*n*(2+n))/168: n in [1..40]]; // Vincenzo Librandi, Mar 24 2014
(PARI) vector(30, n, m=n+1; m^2*(3*m^6 -14*m^4 +21*m^2 -10)/168) \\ G. C. Greubel, Aug 28 2019
(Sage) [n^2*(3*n^6 -14*n^4 +21*n^2 -10)/168 for n in (2..30)] # G. C. Greubel, Aug 28 2019
(GAP) List([2..30], n-> n^2*(3*n^6 -14*n^4 +21*n^2 -10)/168); # G. C. Greubel, Aug 28 2019

Crossrefs

Cf. A000540.

Sequence in context: A223152 A301544 A229328 * A293613 A304838 A337896

Adjacent sequences: A101090 A101091 A101092 * A101094 A101095 A101096

Keyword

nonn,easy

Author

Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 15 2004

Extensions

Edited by Ralf Stephan, Dec 16 2004

Status

approved